Interpretation of a parameter table of NLME analysis

Shen Cheng

2026-02-20

Examples1

Fixed-effect parameters

Table elements1

  • Parameter descriptions

    • Parameter abbreviations with unit: CL/F (L/h)

    • Parameter labels (optional): \(\theta_1\)

    • Parameter descriptions: Apparent central clearance

  • Footnotes:

    • Full name for abbreviations?
    • How were RSE or CI calculated

Table elements1

  • Parameter values:

    • Point estimates

      • Interpretatability: For example, if parameterize \(log(CL)=THETA(1)\) during modeling, reporting \(exp(THETA(1))\) for CL in normal scale.

    • Uncertainty (i.e., precisions or imprecisions)

      • Standard error (SE)

      • Relative standard error (RSE)

      • 95% confidence interval (CI)

Point estimates

  • Output file:.ext
  • Final point estimates in the row with Iteration=-1000000000.

Point estimates (interpretable)

Back-transform model parameters for interpretability

  • For example, if a parameter is modeled in:

    • log scale (log-transformation):

      • Commonly use, natural bound of 0.

      • NONMEM: CL=EXP(THETA(1)+ETA(1)).

      • Back-transform: report \(e^{\theta_1}\) instead of \(\theta_1\).

    • logit scale (logit-transformation):

      • Commonly use, natural bounds between 0 and 1.

      • NONMEM: F=EXP(THETA(1)+ETA(1))/(EXP(THETA(1)+ETA(1))+1).

      • Back-transform: report \(\frac{e^{\theta_1}}{e^{\theta_1}+1}\) instead of \(\theta_1\).

Uncertainty (SE)

  • Output file: .cov
    • SE of a parameter: \(SE=sqrt(diag)\)
    • Although also available in .ext row Iteration=-10000000001.

\[ \begin{bmatrix} \mathbf{x_{11}} & x_{12} & x_{13} & \dots & x_{1n} \\ x_{21} & \mathbf{x_{22}} & x_{23} & \dots & x_{2n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ x_{d1} & x_{d2} & x_{d3} & \dots & \mathbf{x_{dn}} \end{bmatrix} \]

Uncertainty (RSE)

If a parameter is modeled in

  • Normal scale: \(RSE=\frac{SE}{\hat{\theta}} \times 100\%\)1

  • Log scale: \(RSE=sqrt(e^{SE^2}-1)\)

  • Logit scale:

    • \(RSE=\frac{SE}{\hat{\theta}} \times 100\%\) (may not appropriate)

    • \(RSE=\frac{SE^{*}}{\hat{\theta^{*}}} \times 100\%\) (more accurately)

      • \(\hat{\theta^{*}} = \frac{e^{\hat{\theta}}}{e^{\hat{\theta}}+1}\)

      • \(SE^{*}=SE \times \hat{\theta^{*}}\times (1-\hat{\theta^{*}})\) (Delta method)

Uncertainty (confidence interval)

  • Compute CI of transformed parameters:
    • Parametric (symmetrical):
      • Assuming normal distributions: estimates \(\pm\) 1.96*SE.
    • Non-parametric (asymmetrical):
      • Bootstrap, LLP, SIR and Bayesian (week 6).

Random-effect parameters

Table elements

  • Majority of the elements are similar to the fixed-effect parameters:
    • Parameter descriptions, abbreviations, and labels.
    • Point estimates (Iteration=-1000000000 in .ext).
    • Uncertainty: SE, RSE or CV.
    • Footnotes:
      • Full name for abbreviations?
      • How were RSE or CI calculated

Table elements

  • Report \(CV\%\) for BSV terms.
  • Report \(Corr\) for covariance terms.
  • Report \(CV\%\) or \(SD\) for proportional or additive RUV, respectively
  • Report shrinkage.
  • Footnotes: how were \(CV\%\) and \(SD\) of BSV and RUV calculated.

CV% of BSV

  • A log-normal distribution is commonly used for model BSV.
    • Avoid negative values in individual PK parameters (e.g., CL).
    • CL=TVCL* EXP(ETA(1))
    • \(CV\% = sqrt(e^{\omega_{1,1}^2}-1) \times 100\%\)
  • If a normal distribution is used instead
    • CL=TVCL+ETA(1)
    • \(CV\%=sqrt(\omega_{1,1}^2) \times 100\%\)
  • If a logit-normal distribution is used:
    • F=EXP(THETA(1)+ETA(1))/(EXP(THETA(1)+ETA(1))+1)
    • Variability depends on mean.
    • \(CV\%\) is not appropriate, \(SD\) is calculated instead using exact moments1.
# m: mean of the logit term
# v: variance of the logit term
moments <- logitnorm::momentsLogitnorm(mu=m, sigma=sqrt(v))
sd <- sqrt(moments[["var"]])

Corr of covariance terms

  • \(Corr=\frac{COV_{p1,p2}}{sqrt(VAR_{p1}) \times sqrt(VAR_{p2})}\)
    • \(COV_{p1,p2}\): estimated covariance between parameters 1 and 2 (p1 and p2).
    • \(VAR_{p1}\): estimated variance of p1.
    • \(VAR_{p2}\): estimated variance of p2.

\[ \begin{bmatrix} VAR_{p1} & COV_{p1,p2} \\ COV_{p2,p1} & VAR_{p2} \\ \end{bmatrix} \]

  • Example: \(Corr_{(\omega^2_{CL/F},\omega^2_{V2/F})}=\frac{0.0703}{sqrt(0.114) \times sqrt(0.0824)}=0.725\)

CV% or SD for RUV

Combined error model: Y = IPRED*(1+EPS(1))+EPS(2)

  • Report \(CV\%\) for proportional error term: \(CV\%=sqrt(\sigma^2_{1,1})\)
  • Report \(SD\) for additive error term: \(SD=sqrt(\sigma^2_{2,2})\)

Shrinkage (\(\eta\) shrinkage)

  • When the individual data brings only few information (i.e., sparse) about the individual parameter value, the individual estimates (e.g., empirical Bayes estimates, EBEs) is close to (or “shrinks” to) population mean1.
  • Output file:.shk
  • \(\eta_i-shrinkage = 1-\frac{SD_{\eta_i}}{\omega}\) (standard)
  • \(\eta_i-shrinkage = 1-\frac{VAR_{\eta_i}}{\omega^2}\)

\(\eta\) shrinkage1

  • High shrinkage (e.g., SD shrinkage > 30%):
    • Doesn’t necessarily mean the model is unacceptable.
    • Typically occurs when individual data is sparse (e.g., sparse PK sampling).
      • So inadequate information to estimate the EBE of a parameter.
      • EBE distribution may not reflect the actual ones.
    • Diagnostic plots using EBEs are misleading (hiding or suggesting wrong ones):
      • Correlation among \(\eta\)s.
      • Covariate-parameter relationship.
      • DV vs. IPRED

\(\eta\) shrinkage1

Random-effect parameters: Uncertainty (confidence interval)

  • Compute CI of transformed parameters:
    • Parametric (symmetrical):
      • Assuming normal distributions: estimates \(\pm\) 1.96*SE.
    • Non-parametric (asymmetrical):
      • Bootstrap, LLP, SIR and Bayesian (week 6).